Abstract
Quantum bits (qubits) are at the heart of quantum information processing schemes. Currently, solidstate qubits, and in particular the superconducting ones, seem to satisfy the requirements for being the building blocks of viable quantum computers, since they exhibit relatively long coherence times, extremely low dissipation, and scalability. The possibility of achieving quantum coherence in macroscopic circuits comprising Josephson junctions, envisioned by Legett in the 1980’s, was demonstrated for the first time in a charge qubit; since then, the exploitation of macroscopic quantum effects in lowcapacitance Josephson junction circuits allowed for the realization of several kinds of superconducting qubits. Furthermore, coupling between qubits has been successfully achieved that was followed by the construction of multiplequbit logic gates and the implementation of several algorithms. Here it is demonstrated that induced qubit lattice coherence as well as two remarkable quantum coherent optical phenomena, i.e., selfinduced transparency and Dicketype superradiance, may occur during lightpulse propagation in quantum metamaterials comprising superconducting charge qubits. The generated qubit lattice pulse forms a compound ”quantum breather” that propagates in synchrony with the electromagnetic pulse. The experimental confirmation of such effects in superconducting quantum metamaterials may open a new pathway to potentially powerful quantum computing.
Introduction
Quantum simulation, that holds promises of solving particular problems exponentially faster than any classical computer, is a rapidly expanding field of research^{1,2,3}. The information in quantum computers is stored in quantum bits or qubits, which have found several physical realizations; quantum simulators have been nowadays realized and/or proposed that employ trapped ions^{4}, ultracold quantum gases^{5}, photonic systems^{6}, quantum dots^{7}, and superconducting circuits^{1,8,9}. Solid state devices, and in particular those relying on the Josephson effect^{10}, are gaining ground as preferable elementary units (qubits) of quantum simulators since they exhibit relatively long coherence times and extremely low dissipation^{11}. Several variants of Josephson qubits that utilize either charge or flux or phase degrees of freedom have been proposed for implementing a working quantum computer; the recently anounced, commercially available quantum computer with more than 1000 superconducting qubit CPU, known as DWave 2X^{TM} (the upgrade of DWave Two^{TM} with 512 qubits CPU), is clearly a major advancement in this direction. A single superconducting charge qubit (SCQ)^{12} at milikelvin temperatures behaves effectively as an artificial twolevel “atom” in which two states, the ground and the first excited ones, are coherently superposed by Josephson coupling. When coupled to an electromagnetic (EM) vector potential, a single SCQ does behave, with respect to the scattering of EM waves, as an atom in space. Indeed, a “singleatom laser” has been realized with an SCQ coupled to a transmission line resonator (“cavity”)^{13}. Thus, it would be anticipated that a periodic structure of SCQs demonstrates the properties of a transparent material, at least in a particular frequency band. The idea of building materials comprising artificial “atoms” with engineered properties, i.e., metamaterials, and in particular superconducting ones^{14}, is currently under active development. Superconducting quantum metamaterials (SCQMMs) comprising a large number of qubits could hopefully maintain quantum coherence for times long enough to reveal new, exotic collective properties. The first SCQMM that was only recently implemented comprises 20 flux qubits arranged in a double chain geometry^{15}. Furthermore, lasing in the microwave range has been demonstrated theoretically to be triggered in an SCQMM initialized in an easily reachable factorized state^{16}.
Results
Superconducting Quantum Metamaterial Model
Consider an infinite, onedimensional (1D) periodic SCQ array placed in a transmission line (TL) consisting of two superconducting strips of infinite length^{17,18} (Fig. 1a,b); each SCQ, in the form of a tiny superconducting island, is connected to each bank of the TL by a Josephson junction (JJ). The control circuitry for each individual SCQ (Fig. 1c), consisting of a gate voltage source V_{g} coupled to it through a gate capacitor C_{g}, allows for local control of the SCQMM by altering independently the state of each SCQ^{19}. The SCQs exploit the nonlinearity of the Josephson effect and the large charging energy resulting from nanofabrication to create artificial mesoscopic twolevel systems. A propagating EM field in the superconducting TL gives rise to nontrivial interactions between the SCQs, that are mediated by its photons^{20}. Those interactions are of fundamental importance in quantum optics, quantum simulations, and quantum information processing, as well. In what follows, it is demonstrated theoretically that selfinduced transparency^{21} and Dicketype superradiance (collective spontaneous emission)^{22} occur for weak EM fields in that SCQMM structure; the occurence of the former or the latter effect solely depends on the initial state of the SCQ subsystem. Most importantly, selfinduced transparent (SIT) or superradiant (SRD) pulses induce quantum coherence effects in the qubit subsystem. In superradiance (resp. selfinduced transparency), the initial conditions correspond to a state where the SCQs are all in their excited (resp. ground) state; an extended system exhibiting SRD or SIT effects is often called a coherent amlpifier or attenuator, respectively. These fundamental quantum coherent prosesses have been investigated extensively in connection to one and twophoton resonant twolevel systems. Superradiant effects have been actually observed recently in twolevel systems formed by quantum dot arrays^{23} and spinorbit coupled BoseEinstein condensates^{24}; the latter system features the coupling between momentum states and the collective atomic spin which is analogous to that between the EM field and the atomic spin in the original Dicke model. These results suggest that quantum dots and the atoms in the BoseEinstein condensate can radiatively interact over long distances. The experimental confirmation of SIT and SRD in extended SCQMM structures may open a new pathway to potentially powerful quantum computing. As a consequence of these effects, the value of the speed of either an SIT or SRD propagating pulse in a SCQMM structure can in principle be engineered through the SCQ parameters^{25}, which is not possible in ordinary resonant media. From a technological viewpoint, an EM (light) pulse can be regarded as a “bit” of optical information; its slowing down, or even its complete halting for a certain time interval, may be used for data storage in a quantum computer.
In the following, the essential building blocks of the SCQMM model are summarized in a selfcontained manner, yet omitting unnecessary calculational details which are presented in the Supplementary Information. The energy per unit cell of the SCQMM structure lying along the x–direction, when coupled to an EM vector potential , can be readily written as^{17,18}
in units of the Josephson energy E_{J} = Φ_{0}I_{c}/(2πC), with Φ_{0}, I_{c} and C being the magnetic flux quantum, the critical current of the JJ, and the capacitance of the JJ, respectively. In equation (1), φ_{n} is the superconducting phase on the nth island, β = (8πdE_{J})^{−1/2}(Φ_{0}/2π), with d being the separation between the electrodes of the superconducting TL, and the overdots denote differentiation with respect to the temporal variable t. Assuming EM fields with wavelengths , with being the distance between neighboring qubits, the EM potential is approximately constant within a unit cell, so that in the centre of the nth unit cell . In terms of the discretized EM potential A_{z,n}(t), the normalized gauge term is a_{n} = 2πdA_{x,n}/Φ_{0}. The classical energy expression equation (1) provides a minimal modelling approach for the system under consideration; the three angular brackets in that equation correspond to the energies of the SCQ subsystem, the EM field inside the TL electrodes, and their interaction, respectively. The latter results from the requirement for gaugeinvariance of each Josephson phase.
Second Quantization and Reduction to MaxwellBloch Equations
The quantization of the SCQ subsystem requires the replacement of the classical variables φ_{n} and by the corresponding quantum operators and , respectively. While the EM field is treated classically, the SCQs are regarded as twolevel systems, so that only the two lowest energy states are retained; under these considerations, the secondquantized Hamiltonian corresponding to equation (1) is
where p, p′ = 0, 1, E_{0} and E_{1} are the energy eigenvalues of the ground and the excited state, respectively, the operator (a_{n,p}) excites (deexcites) the nth SCQ from the ground to the excited (from the excited to the ground) state, and are the matrix elements of the effective SCQEM field interaction. The basis states Ξ_{p} can be obtained by solving the singleSCQ Schrödinger equation (−∂^{2}/∂φ^{2} − E_{p} + 2 cos φ)Ξ_{p} = 0. In general, each SCQ is in a superposition state of the form . The substitution of Ψ_{n}〉 into the Schrödinger equation with the secondquantized Hamiltonian equation (2), and the introduction of the Bloch variables , , R_{z}(n) = Ψ_{n,1}^{2} − Ψ_{n,0}^{2}, provides the reformulation of the problem into the MaxwellBloch (MB) equations
that are nonlinearly coupled to the resulting EM vector potential equation
where δα_{n} = α_{n−1} − 2α_{n} + α_{n+1}, D = (V_{11} − V_{00})/(2χ), Ω^{2} = (V_{00} + V_{11})/2, μ = V_{10}/χ = V_{01}/χ, and Δ = ε_{1} − ε_{0} ≡ (E_{1} − E_{0})/χ, with χ = ω_{J}/E_{J}. In the earlier equations, the overdots denote differentiation with respect to the normalized time t → ω_{J}t, in which ω_{J} = eI_{c}/(C) is the Josephson frequency and e, are the electron charge and the Planck’s constant devided by 2π, respectively.
Approximations and Analytical Solutions
For weak EM fields, the approximation can be safely used. Then, by taking the continuum limit α_{n}(t) → α(x, t) and R_{i}(n; t) → R_{i}(x; t) (i = x, y, z) of equations (3–, , 6), a set of simplified, yet still nonlinearly coupled equations is obtained, similar to those encountered in twophoton SIT in resonant media^{27}. Further simplification can be achieved with the slowly varying envelope approximation (SVEA) by making for the EM vector potential the ansatz α(x, t) = ε(x, t)cos Ψ(x, t), where Ψ(x, t) = kx − ωt + ϕ(x, t) and ε(x, t), ϕ(x, t) are the slowly varying pulse envelope and phase, respectively, with ω and being the frequency of the carrier wave of the EM pulse and its wavenumber in the superconducting TL, respectively. In the absence of the SCQ chain the EM pulse is “free” to propagate in the TL with speed β. At the same time, equations (3, 4, 5) for the Bloch vector components are transformed according to R_{x} = r_{x} cos (2Ψ) + r_{y} sin (2Ψ), R_{y} = r_{y} cos (2Ψ) − r_{x} sin (2Ψ), and R_{z} = r_{z}. Then, collecting the coefficients of sinΨ and cosΨ while neglecting the rapidly varying terms, and averaging over the phase Ψ, results in a set of truncated equations (see Supplementary Information). Further manipulation of the resulting equations and the enforcement of the twophoton resonance condition Δ = 2ω, results in
where c = β^{2}k/ω = 2β^{2}k/Δ, and the truncated MB equations
which obey the conservation law . In equation (9), the n–dependence of the r_{i} (i = x, y, z) is suppressed, in accordance with common practices in quantum optics.
The r_{i} can be written in terms of new Bloch vector components S_{i} using the unitary transformation r_{x} = S_{x} cos Φ − S_{z} sin Φ, r_{y} = S_{y}, and r_{z} = S_{z} cos Φ + S_{x} sin Φ, where Φ is a constant angle to be determined. Using a procedure similar to that for obtaining the r_{i}, we get , , and , where and tan Φ ≡ γ = 4D/μ. The combined system of the equations for the S_{i} and equations (7, 8) admits exact solutions of the form ε = ε(τ = t − x/v) and S_{i} = S_{i}(τ = t − x/v), where v is the pulse speed. For the slowly varying pulse envelop, we obtain
where is the pulse amplitude and τ_{p} = {χ(σμ/ω)[v/(c − v)]}^{−1} its duration, with . The decoherence factor γ can be expressed as a function of the matrix elements of the SCQEM field interaction, V_{ij}, as γ = 2(V_{11} − V_{00})/V_{10} that can be calculated when the latter are known. Such Lorentzian propagating pulses have been obtained before in twophoton resonant media^{28,29}; however, SIT in quantum systems has only been demonstrated in onephoton (absorbing) frequency gap media, in which solitonic pulses can propagate without dissipation^{30}. The corresponding solution for the population inversion, R_{z}, reads
where , and the plus (minus) sign corresponds to absorbing (amplifying) SCQMMs; these are specified through the initial conditions as R_{z}(−∞) = −1, ε(−∞) = 0 and R_{z}(−∞ = +1), ε(−∞) = 0 for absorbing and amplifying SCQMMs, respectively (with R_{x}(−∞) = R_{y}(−∞) = 0 in both cases). The requirement for the wavenumber k being real, leads to the SCQ parameterdependent condition 2χ^{2}(V_{11} + V_{00}) < (E_{1}−E_{0})^{2} for pulse propagation in the SCQMM. Thus, beyond the obtained twophoton SIT or SRD, the propagating EM pulse plays a key role in the interaction processes in the qubit subsystem: it leads to collective behavior of the ensemble of SCQs in the form of quantum coherent probability pulses; such pulses are illustrated here through the population inversion R_{z}.
The corresponding velocityamplitude relation of the propagating pulse reads
Equation (12) can be also written as a velocityduration expression, since the pulse amplitude and its duration are related through . The duration of SRD pulses cannot exceed the limiting value of τ_{M} = ω(c − v)/(χμv). From equation (12), the existence of a critical velocity c, defined earlier, can be immediately identified; that velocity sets an upper (lower) bound on the pulse velocity in absorbing (amplifying) SCQMM structures. Thus, in absorbing (amplifying) SCQMM structures, pulses of higher intensity propagate faster (slower). That limiting velocity is generally lower than the corresponding one for twophoton SIT or SRD in ordinary media, β, which here coincides with the speed of the “free” pulse in the TL (Fig. 2). As can be inferred from Fig. 2, the increase of decoherence through γ makes the velocity to saturate at its limiting value c at lower amplitudes ε; that velocity can be reduced further with increasing the ratio of the TL to the pulse carrier wave frequency Ω/ω through proper parameter engineering. Moreover, effective control of v in SCQMMs could in principle be achieved by an external field^{31} or by real time tuning of the qubit parameters. That ability to control the flow of “optical”, in the broad sense, information may have technological relevance to quantum computing^{25}. Note that total inversion, i.e. excitation or deexcitation of all qubits during pulse propagation is possible only if γ = 0, i.e., for V_{00} = V_{11}; otherwise (V_{00} < V_{11}) the energy levels of the qubit states are Starkshifted, violating thus the resonance condition. Typical analytical profiles for the EM vector potential pulse ε(τ) and the population inversions R_{z}(τ) both for absorbing and amplifying SCQMMs are shown in the insets of Fig. 2. The maximum of ε(τ) reduces considerably with increasing γ, while at the same time the maximum (minimum) of R_{z} decreases (increases) at the same rate.
The system of equations (7, –9) can be reduced to a single equation using the parametrization r_{x} = R_{0}γσ^{2}[1 − cos θ], r_{y} = −R_{0}σ sin θ, and r_{z} = R_{0}{1 − σ^{2}[1 − cos θ]}, of the Bloch vector components. Then, a relation between the Bloch angle θ ≡ θ(x, t) and the slow amplitude ε can be easily obtained, that leads straightforwardly to the equation . Time integration of that equation yields , that conforms with the famous area theorem: pulses with special values of “area” θ(x) = 2πn conserve that value during propagation.
Here we concentrate on the interaction of the SCQs with the EM wave and we are not concerned with decoherence effects in the SCQs due to dephasing and energy relaxation. This is clearly an idealization which is justified as long as the coherence time exceeds the wave propagation time across a relatively large number of unit cell periods. In a recent experiment^{26}, a charge qubit coupled to a strip line had a dephasing time in excess of 200 ns, i.e., a dephasing rate of 5 MHz, and a photon loss rate from the cavity of 0.57 MHz. Those frequencies are very small compared with the transition frequency of the considered SCQs which is of the order of the Josephson energy (i.e., a few GHz)^{17,18}. Therefore, we have neglected such decoherence effects in the present work. The decoherence factor γ, which in Fig. 2b,d has been chosen according to the parameter values in ref. 17, is not related to either dephasing or energy relaxation. That factor attains a nonzero value whenever the matrix elements of the effective SCQEM field interaction, V_{11} and V_{00}, are not equal.
Numerical Simulations
In order to confirm numerically the obtained results, the equations (3–, , 6) are integrated in time using a fourth order RungeKutta algorithm with constant timestep. For pulse propagation in absorbing SCQMMs, all the qubits are initially set to their ground state while the vector potential pulse assumes its analytical form for the given set of parameters. A very fine timestep and very large qubit arrays are used to diminish the energy and/or probability loss and the effects of the boundaries during propagation, respectively. The subsequent temporal evolution in twophoton SIT SCQMM, as can be seen in Fig. 3a,b, in which several snapshots of the population inversion R_{z}(n; t) and the vector potential pulses a_{n}(t), respectively, are shown, reveals that the latter are indeed capable of inducing quantum coherent effects in the qubit subsystem in the form of population inversion pulses! In Fig. 3a, the amplitude of the R_{z}(n; t) pulse gradually grow to the expected maximum around unity in approximately 60 time units, and they continue its course almost coherently (although with fluctuating amplitude) for about 160 more time units, during which they move at the same speed as the vector potential pulse (Fig. 3b). However, due to the inherent discreteness in the qubit subsystem and the lack of interqubit coupling, the R_{z}(n; t) pulse splits at certain instants leaving behind small “probability bumps” that get pinned at particular qubits. After the end of the almost coherent propagation regime, the R_{z}(n; t) pulse broadens and slowsdown until it stops completely. At the same time, the width of the a_{n}(t) pulse increases in the course of time due to discretenessinduced dispersion. A comparison with the corresponding analytical expressions reveals fair agreement during the almost coherent propagation regime, although both the R_{z}(n; t) and a_{n}(t) pulses travel slightly faster than expected from the analytical predictions. The temporal variable here is normalized to the inverse of the Josephson frequency ω_{J} which for typical parameter values is of the order of a few GHz^{17}. Then, the almost coherent induced pulse regime in the particular case shown in Fig. 3 lasts for ~160 × 10^{−9} s, or ~160 ns, which is of the same order as the reported decoherence time for a charge qubit in ref. 26 (i.e., 200 ns).
The situation seems to be different, however, in the case of twophoton SRD pulses, as can be observed in the snapshots shown in Fig. 3c,d for R_{z}(n; t) and a_{n}(t), respectively. Here, the lack of the interqubit interaction is crucial, since the SCQs that make a transition from the excited to their ground state as the peak of the a_{n}(t) pulse passes by their location, cannot return to their excited states after the a_{n}(t) pulse has gone away. It seems, thus, that the a_{n}(t) pulse creates a type of a kinklike front that propagates at the same velocity. It should be noted that the common velocity of the R_{z}(n; t) and a_{n}(t) pulses is considerably different (i.e., smaller) than the analytically predicted one, as it can be inferred by inspection of Fig. 3c,d. Even more complicated behavioral patterns of twophoton SRD propagating pulses and the effect of nonzero decoherence factor are discussed in the Supplementary Information.
Conclusion
An SCQMM comprising SCQs loaded periodically on a superconducting TL has been investigated theoretically using a minimalistic onedimensional model following a semiclassical approach. While the SCQs are regarded as twolevel quantum systems, the EM field is treated classically. Through analytical techniques it is demonstrated that the system allows selfinduced transparent and superradiant pulse propagation given that a particular constraint is fulfilled. Most importanty, it is demonstrated that the propagating EM pulses may induce quantum coherent population inversion pulses in the SCQMM. Numerical simulation of the semiclassical equations confirms the excitation of population inversion pulses with significant coherence time in absorbing media. The situation is slightly different in amplifying media, in which the numerically obtained, induced population inversion excitations are kinklike propagating structures (although more complex behaviors discussed in the Supplementary Information also appear). Moreover, the limiting pulse velocity in both amplifying and absorbing SCQMMs is lower than the corresponding one in twophoton resonant amplifying and absorbing ordinanary (atomic) media. That limiting velocity in SCQMMs can in principle be engineered through the SCQ parameters.
Additional Information
How to cite this article: Ivić, Z. et al. Qubit lattice coherence induced by electromagnetic pulses in superconducting metamaterials. Sci. Rep. 6, 29374; doi: 10.1038/srep29374 (2016).
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Acknowledgements
This work was partially supported by the European Union Seventh Framework Programme (FP7REGPOT201220131) under grant agreement n^{o} 316165, the Serbian Ministry of Education and Science under Grants No. III–45010, No. OI–171009, the Ministry of Education and Science of the Republic of Kazakhstan (Contract No. 339/762015), and the Ministry of Education and Science of the Russian Federation in the framework of the Increase Competitiveness Program of NUST “MISiS”(No. K22015007).
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Z.I., N.L. and G.P.T. performed the research, analyzed the results and wrote the paper.
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Ivić, Z., Lazarides, N. & Tsironis, G. Qubit lattice coherence induced by electromagnetic pulses in superconducting metamaterials. Sci Rep 6, 29374 (2016). https://doi.org/10.1038/srep29374
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